## Table of Contents

## Prefixes

Prefixes are useful for expressing units of physical quantities that are either very big or very small. If you write out such big or small numbers, it will be time consuming and prone to errors.

- In the English language, prefixes are letters which are placed in front of words to make new words.

Some of the Greek prefixes and their symbols to indicate decimal sub-multiples and multiples of the SI units are:

Number | Prefix | Number | Prefix |
---|---|---|---|

$10^{-9}$ | nano (n) | $10^{-1}$ | deci (d) |

$10^{-6}$ | micro ($\mu$) | $10^{3}$ | Kilo (K) |

$10^{-3}$ | milli (m) | $10^{6}$ | Mega (M) |

$10^{-2}$ | centi (c) | $10^{9}$ | Giga (G) |

## Common Examples Of Prefixes

- 1 cm = 0.01 metre
- 1 kg = 1000 grams
- 1 GB = 1 000 000 000 bytes
- 0.000 0052 s = $5.2 \, \mu \text{s}$ = $5.2 \times 10^{-6} \, \text{s}$

## Standard Form & Order Of Magnitude

$5.2 \times 10^{-6} \, \text{s}$ is what is known as **standard form**. When numbers are too large or too small, it is convenient to express them in standard form in the following format:

$$\text{M} \times 10^{\text{N}}$$

, where:

$\text{M}$ is in the range of: $1 \leq \text{M} < 10$

$\text{N}$ denotes the **order of magnitude** and is an integer.

Order of magnitude are used to estimate numbers which are extremely large to the nearest power of ten. The table below shows how orders of magnitude are used to compare base quantities – mass and length.

Mass ($\text{kg}$) | Factor | Length ($\text{m}$) | Factor |
---|---|---|---|

Electron | $10^{-30}$ | Radius of proton | $10^{-15}$ |

Proton | $10^{-27}$ | Radius of atom | $10^{-10}$ |

Ant | $10^{-3}$ | Height of ant | $10^{-3}$ |

Human | $10^{1}$ | Height of human | $10^{0} = 1$ |

Earth | $10^{24}$ | Radius of Earth | $10^{7}$ |

Sun | $10^{30}$ | Radius of Sun | $10^{9}$ |

Order of magnitude is useful when you do “back-of-the-envelope” calculations. A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope.

Back-of-the-envelope calculations are used to quickly check something. If your back-of-the-envelope calculations yield several orders of magnitude bigger or smaller than what you expect, your formula or input variables must be wrong.

## Worked Examples

### Example 1

The length of the world’s smallest playable guitar is 13 micrometers ($13 \, \mu \text{m}$). Represent the guitar’s length in standard form.

**Click here to show/hide answer**

$$\begin{aligned} 13 \, \mu \text{m} &= 0.000013 \text{ m} \\ &= 13 \times 10^{-6} \text{ m} \\ &= 1.3 \times 10^{-5} \text{ m} \end{aligned}$$

The guitar’s length is $1.3 \times 10^{-5} \text{ m}$ in standard form.

### Example 2

What level of precision is implied in an order-of-magnitude calculation?

**Click here to show/hide answer**

Zero significant digits. An order-of-magnitude calculation is accurate only within a factor of 10.